Interpret alpha's on Dual-Beta Model regression Results

I am trying to calculate the Dual-Beta for Apple (AAPL) by running a regression against the Spyder's ETF (SPY) & using the 10-yr Risk Free rate. The formula for the dual beta is:

($r_{AAPL}-r_f) = \alpha^+D + \beta^+( r^+_m - r_f )+\alpha^-(1-D) + \beta^-( r^-_m - r_f )$

and can be found here: http://en.wikipedia.org/wiki/Dual-beta

I downloaded AAPL & SPY data from Yahoo Finance and the 10-Year Rate from FRED and put them in Excel. The dates are from 2013-10-31 to 2014-10-31.

[AAPL-Rf]   [R+]    [R-]    [Dummy+]
-0.01%  -0.01%  0.00%   1
-0.52%  0.23%   0.00%   1
1.28%   0.35%   0.00%   1
-0.25%  0.00%   -0.32%  0
-0.29%  0.50%   0.00%   1
-1.65%  0.00%   -1.27%  0
1.56%   1.34%   0.00%   1
-0.30%  0.01%   0.00%   1
0.17%   0.00%   -0.21%  0
0.12%   0.80%   0.00%   1
1.43%   0.49%   0.00%   1
-0.62%  0.43%   0.00%   1
-1.22%  0.00%   -0.36%  0
0.17%   0.00%   -0.23%  0
-0.89%  0.00%   -0.32%  0
1.17%   0.80%   0.00%   1
-0.27%  0.49%   0.00%   1
0.76%   0.00%   -0.11%  0
1.81%   0.02%   0.00%   1
2.33%   0.24%   0.00%   1
1.83%   0.00%   -0.07%  0
-0.88%  0.00%   -0.27%  0
2.69%   0.00%   -0.44%  0
-0.25%  0.00%   -0.02%  0
0.51%   0.00%   -0.44%  0
-1.41%  1.11%   0.00%   1
1.13%   0.25%   0.00%   1
-0.16%  0.00%   -0.37%  0
-0.75%  0.00%   -1.13%  0
-0.15%  0.00%   -0.34%  0
-1.11%  0.00%   -0.02%  0
0.54%   0.62%   0.00%   1
-0.46%  0.00%   -0.33%  0
-0.78%  1.70%   0.00%   1
-1.15%  0.00%   -0.12%  0
0.83%   0.57%   0.00%   1
3.76%   0.53%   0.00%   1
-0.43%  0.21%   0.00%   1
-0.67%  0.50%   0.00%   1
-0.69%  0.00%   -0.01%  0
-1.00%  0.00%   -0.02%  0
1.15%   0.47%   0.00%   1
-1.41%  0.00%   -0.97%  0
-2.23%  0.00%   -0.02%  0
0.53%   0.00%   -0.30%  0
-0.72%  0.61%   0.00%   1
0.62%   0.01%   0.00%   1
-1.30%  0.05%   0.00%   1
-0.67%  0.27%   0.00%   1
0.51%   0.00%   -1.34%  0
1.97%   1.08%   0.00%   1
1.98%   0.53%   0.00%   1
-0.57%  0.00%   -0.14%  0
-2.49%  0.00%   -0.43%  0
1.53%   0.29%   0.00%   1
0.44%   0.06%   0.00%   1
0.83%   0.00%   -0.83%  0
-1.84%  0.00%   -2.14%  0
0.81%   0.00%   -0.50%  0
-8.35%  0.59%   0.00%   1
-1.14%  0.00%   -0.97%  0
-0.21%  1.06%   0.00%   1
0.15%   0.00%   -0.60%  0
0.18%   0.00%   -2.26%  0
1.43%   0.70%   0.00%   1
0.75%   0.00%   -0.13%  0
0.57%   1.31%   0.00%   1
1.38%   1.23%   0.00%   1
1.77%   0.17%   0.00%   1
1.29%   1.09%   0.00%   1
-0.01%  0.04%   0.00%   1
1.56%   0.50%   0.00%   1
-0.09%  0.55%   0.00%   1
0.36%   0.11%   0.00%   1
-1.60%  0.00%   -0.67%  0
-1.17%  0.59%   0.00%   1
-1.12%  0.00%   -0.12%  0
0.42%   0.55%   0.00%   1
-1.05%  0.00%   -0.05%  0
-0.92%  0.00%   0.00%   1
1.97%   0.51%   0.00%   1
-0.28%  0.25%   0.00%   1
0.27%   0.00%   -0.71%  0
0.66%   1.40%   0.00%   1
0.19%   0.08%   0.00%   1
-0.30%  0.22%   0.00%   1
-0.07%  0.04%   0.00%   1
0.09%   0.00%   -0.06%  0
0.96%   0.00%   -0.50%  0
0.10%   0.01%   0.00%   1
-1.13%  0.00%   -1.13%  0
-1.13%  0.00%   -0.29%  0
0.38%   0.90%   0.00%   1
0.88%   0.71%   0.00%   1
-0.03%  0.00%   -0.54%  0
-0.49%  0.58%   0.00%   1
0.78%   0.00%   -0.40%  0
1.17%   0.00%   -0.42%  0
1.06%   0.47%   0.00%   1
-0.97%  0.00%   -0.73%  0
-0.43%  0.00%   -0.21%  0
-0.13%  0.48%   0.00%   1
-0.03%  0.81%   0.00%   1
0.91%   0.66%   0.00%   1
0.16%   0.32%   0.00%   1
-0.71%  0.00%   -0.14%  0
-1.30%  0.00%   -1.19%  0
-1.59%  0.00%   -1.11%  0
-0.02%  0.40%   0.00%   1
1.31%   1.07%   0.00%   1
-1.31%  0.00%   -2.11%  0
-0.75%  0.00%   -0.91%  0
0.39%   0.78%   0.00%   1
-0.71%  0.68%   0.00%   1
0.20%   1.04%   0.00%   1
1.13%   0.13%   0.00%   1
1.17%   0.34%   0.00%   1
0.09%   0.45%   0.00%   1
-1.32%  0.00%   -0.24%  0
7.87%   0.19%   0.00%   1
0.73%   0.00%   -0.82%  0
3.79%   0.31%   0.00%   1
-0.31%  0.46%   0.00%   1
-0.38%  0.29%   0.00%   1
0.22%   0.00%   0.00%   1
0.18%   0.00%   -0.15%  0
1.39%   0.19%   0.00%   1
-1.10%  0.00%   -0.88%  0
-0.36%  0.58%   0.00%   1
-0.19%  0.00%   -0.11%  0
-0.43%  0.14%   0.00%   1
1.23%   0.97%   0.00%   1
0.15%   0.08%   0.00%   1
0.02%   0.00%   -0.48%  0
-0.86%  0.00%   -0.88%  0
1.46%   0.34%   0.00%   1
1.16%   0.36%   0.00%   1
0.02%   0.00%   -0.64%  0
0.26%   0.83%   0.00%   1
0.14%   0.24%   0.00%   1
1.12%   0.39%   0.00%   1
1.84%   0.61%   0.00%   1
-0.27%  0.00%   -0.08%  0
1.80%   0.51%   0.00%   1
-0.38%  0.16%   0.00%   1
-0.70%  0.11%   0.00%   1
1.40%   0.00%   -0.06%  0
1.12%   0.19%   0.00%   1
0.38%   0.65%   0.00%   1
-0.28%  0.47%   0.00%   1
1.58%   0.10%   0.00%   1
0.57%   0.00%   0.00%   1
-0.41%  0.00%   -0.35%  0
-1.70%  0.00%   -0.72%  0
-1.10%  0.30%   0.00%   1
0.99%   0.07%   0.00%   1
-0.14%  0.27%   0.00%   1
0.10%   0.73%   0.00%   1
-0.35%  0.10%   0.00%   1
-1.05%  0.20%   0.00%   1
-0.10%  0.00%   -0.04%  0
-0.62%  0.00%   -0.61%  0
0.08%   0.45%   0.00%   1
0.59%   0.00%   -0.08%  0
1.17%   0.19%   0.00%   1
1.03%   0.00%   -0.06%  0
0.63%   0.66%   0.00%   1
-0.05%  0.09%   0.00%   1
0.57%   0.49%   0.00%   1
2.05%   0.00%   -0.36%  0
-0.66%  0.00%   -0.65%  0
0.04%   0.44%   0.00%   1
-0.38%  0.00%   -0.40%  0
0.18%   0.13%   0.00%   1
1.27%   0.49%   0.00%   1
-1.18%  0.00%   -0.20%  0
-0.58%  0.36%   0.00%   1
-1.80%  0.00%   -1.14%  0
1.42%   1.01%   0.00%   1
-0.52%  0.00%   -0.19%  0
0.81%   0.43%   0.00%   1
2.57%   0.22%   0.00%   1
-0.17%  0.00%   0.00%   1
0.65%   0.00%   -0.48%  0
1.36%   0.03%   0.00%   1
-0.66%  0.00%   -0.44%  0
-0.24%  0.01%   0.00%   1
-2.63%  0.00%   -1.98%  0
0.54%   0.00%   -0.31%  0
-0.56%  0.72%   0.00%   1
-0.50%  0.00%   -0.98%  0
-0.18%  0.02%   0.00%   1
-0.02%  0.00%   -0.55%  0
0.27%   1.15%   0.00%   1
1.30%   0.28%   0.00%   1
-0.03%  0.00%   -0.15%  0
1.31%   0.67%   0.00%   1
0.26%   0.47%   0.00%   1
0.48%   0.00%   -0.03%  0
1.19%   0.83%   0.00%   1
1.37%   0.52%   0.00%   1
0.03%   0.26%   0.00%   1
0.00%   0.29%   0.00%   1
0.73%   0.00%   -0.16%  0
0.21%   0.50%   0.00%   1
-0.65%  0.06%   0.00%   1
1.22%   0.00%   -0.05%  0
0.11%   0.00%   -0.06%  0
0.24%   0.28%   0.00%   1
0.77%   0.00%   -0.06%  0
-4.32%  0.00%   -0.06%  0
-0.84%  0.00%   -0.15%  0
0.86%   0.44%   0.00%   1
-0.63%  0.00%   -0.26%  0
-0.38%  0.00%   -0.64%  0
3.02%   0.37%   0.00%   1
0.42%   0.11%   0.00%   1
0.22%   0.00%   -0.59%  0
-0.04%  0.00%   -0.08%  0
-0.77%  0.75%   0.00%   1
0.70%   0.13%   0.00%   1
0.20%   0.52%   0.00%   1
-0.83%  0.00%   -0.10%  0
0.09%   0.00%   -0.78%  0
1.54%   0.00%   -0.58%  0
-0.88%  0.78%   0.00%   1
-3.89%  0.00%   -1.62%  0
2.89%   0.79%   0.00%   1
-0.64%  0.00%   -0.19%  0
0.63%   0.00%   -0.27%  0
-1.58%  0.00%   -1.36%  0
0.72%   0.01%   0.00%   1
-0.29%  1.09%   0.00%   1
-0.01%  0.00%   -0.12%  0
-0.88%  0.00%   -1.55%  0
2.05%   1.74%   0.00%   1
0.21%   0.00%   -1.99%  0
-0.29%  0.00%   -1.15%  0
-0.92%  0.00%   -1.65%  0
-1.07%  0.15%   0.00%   1
-1.24%  0.00%   -0.68%  0
-1.33%  0.00%   -0.09%  0
1.45%   1.17%   0.00%   1
2.11%   0.96%   0.00%   1
2.67%   1.97%   0.00%   1
0.50%   0.00%   -0.72%  0
1.76%   1.16%   0.00%   1
0.37%   0.76%   0.00%   1
-0.11%  0.00%   -0.14%  0
1.53%   1.14%   0.00%   1
0.55%   0.00%   -0.16%  0
-0.34%  0.63%   0.00%   1


For the regression AAPL-Rf is the dependent variable and R+ is the Up-Beta market risk premium, R- is the Down-Beta risk premium , and Dummy+ is 1 when SPY was positive or unchanged, and 0 otherwise (or negative).

The regression results came back as:

AAPL-Rf =- 0.0003 + 0.4059 * [R+] + 0.6111 * [R-] + 0.0036 * [Dummy+]

I am a bit confused on the alpha's... The model describes that positive alpha is multiplied by the Dummy Variable which in this case its: 0.0036 * [Dummy+] but what about the downside alpha? would that be: - 0.0003 ?

Any input would be helpful....

1 Answer

Yes, the downside alpha is -0.0003. You can confirm that result if you include (1-D) explicitly as one more covariate, but then you'll have to request a model with no intercept.

• got it @James . thanks for helping me out – Jason Guevara Nov 5 '14 at 7:35